A Deep Latent Gaussian Model (DLGM) [1]} is a directed graphical model with Gaussian latent variables that are stacked in \(L\)  stochastic layers and that are transformed at each layer by a differentiable nonlinear function parameterized by a neural network \(\mathbf {NN}(\mathbf {z};\theta )\)  consisting of \(K^{(l)}\)  deterministic hidden layers.
Whereas the latent variables in the first DLGMs such as the Variational Autoencoder (VAE) [2]} are assumed to be independent, recent extensions [3]}, [4]} have introduced a deep latent hierarchy that endows the latent space with complex rich structure that can capture complex patterns of dependencies between the latent variables. A general DLGM model has the following generative formulation.
\(\begin{split}\mathbf {z}_n^{(L)}\ &\sim \ \mathcal {N}_{d_{(L)}}\left(0,\, \mathbf {I}\right) \quad n=1, \cdots ,N \\\mathbf {z}_n^{(l)}\ &\sim \ \mathcal {N}_{d_{(l)}}\Big (\mathbf {z}_n^{(l)} \mid \mathbf {NN}^{(l)}(\mathbf {z}_n^{(l+1)};\theta ^{(l)}),\, \Sigma ^{(l)} \Big ) \quad l=1, \cdots ,L-1\\\mathbf {y}_n \mid \mathbf {z}_n &\sim \ \mathbf {Expon}\Big (\mathbf {y}_n \mid \mathbf {NN}^{(0)}(\mathbf {z}_n^{(1)};\theta ^{(0)})\Big )\end{split}\) 
